PHYSICS · LESSON 01 OF 9
Units and measurement
Physics is measured quantities with units. Convert carefully, check dimensions, and report results with sensible precision.
What this lesson explains
A number without a unit means nothing in physics: “the speed is 25” could be 25 m/s or 25 km/h, a factor of 3.6 apart. Many real engineering failures came from mixed units. Units are also your best error detector: if an equation for a distance produces m/s, it is wrong before you compare any numbers.
Every experiment also has limited precision. Knowing how many digits are meaningful stops you reporting 3.14159 m for a length measured with a ruler.
Before you begin
Powers of ten
10a × 10b = 10a+b; 10a / 10b = 10a−b. Scientific notation writes 0.00052 as 5.2 × 10−4.
Rearranging a formula
To make t the subject of v = at, divide both sides by a: t = v/a. Do the same operation to both sides.
The idea, made visible
Every physical quantity is a number times a unit. The SI system uses seven base units; in mechanics you need three: the metre (m) for length, the kilogram (kg) for mass, and the second (s) for time. All other mechanical units are combinations: speed in m/s, acceleration in m/s2, force in newtons, 1 N = 1 kg·m/s2, energy in joules, 1 J = 1 N·m.
To convert units, multiply by conversion factors that equal 1, such as 1000 m / 1 km or 1 h / 3600 s. Arrange each factor so the unwanted unit cancels, exactly like cancelling numbers in a fraction. Because each factor equals 1, the physical quantity does not change — only its description does.
Prefixes scale units by powers of ten: kilo (k) = 103, centi (c) = 10−2, milli (m) = 10−3, micro (μ) = 10−6. For squared and cubed units the factor is squared or cubed: 1 m2 = (100 cm)2 = 104 cm2, and 1 m3 = 106 cm3 = 1000 L.
Dimensional analysis checks equations: both sides, and every term added together, must have the same dimensions. In x = x0 + v0t + 1 / 2at2, each term is a length: m, (m/s)(s) = m, (m/s2)(s2) = m. You can add metres to metres, never metres to seconds.
Significant figures communicate precision. When multiplying or dividing, keep as many significant figures as the least precise input. When adding or subtracting, keep as many decimal places as the least precise input. Keep extra digits during a calculation and round only the final answer.
| prefix | symbol | factor | example |
|---|---|---|---|
| giga | G | 10⁹ | 1 GW = 10⁹ W |
| mega | M | 10⁶ | 1 MPa = 10⁶ Pa |
| kilo | k | 10³ | 1 km = 1000 m |
| centi | c | 10⁻² | 1 cm = 0.01 m |
| milli | m | 10⁻³ | 1 mm = 0.001 m |
| micro | μ | 10⁻⁶ | 1 μs = 10⁻⁶ s |
| nano | n | 10⁻⁹ | 1 nm = 10⁻⁹ m |
Note the capital M (mega) versus lower-case m (milli): a factor of 10⁹ apart.
Key terms
- SI base units (mechanics)
- metre (m), kilogram (kg), second (s). Others include the kelvin (K), mole (mol) and ampere (A).
- Derived unit
- A combination of base units, e.g. newton N = kg·m/s2, joule J = N·m = kg·m2/s2, watt W = J/s, pascal Pa = N/m2.
- Conversion factor
- A ratio equal to 1, such as 1000 m/1 km, used to change units without changing the quantity.
- Dimension
- The type of a quantity: length [L], mass [M], time [T]. Speed has dimensions [L]/[T].
- Significant figures
- The digits that carry meaning about precision: all non-zero digits, zeros between them, and trailing zeros after a decimal point. Leading zeros (0.004) are not significant.
- Accuracy and precision
- Accuracy: closeness to the true value. Precision: closeness of repeated measurements to each other (and the fineness of the scale).
The formulas and what they mean
| Symbol | Meaning | Unit |
|---|---|---|
| m, kg, s | metre, kilogram, second | SI base units |
| N | newton, unit of force | kg·m/s² |
| J | joule, unit of energy and work | N·m = kg·m²/s² |
| ρ (rho) | density = mass/volume | kg/m³ (or g/cm³) |
Speed conversion
1 m/s = 3.6 km/h; v(m/s) = v(km/h) / 3.6
Conditions and limits: Exact, since 1 km = 1000 m and 1 h = 3600 s.
Area and volume conversions
1 m2 = 104 cm2; 1 m3 = 106 cm3 = 103 L; 1 mL = 1 cm3
Conditions and limits: Square or cube the length factor.
Density
ρ = m / V
Conditions and limits: For a uniform material. 1 g/cm3 = 1000 kg/m3 (water ≈ 1.00 g/cm³ near room temperature).
Dimensional consistency
[left side] = [right side]; only like dimensions can be added
Conditions and limits: Always required. Consistency does not prove an equation correct (pure-number factors such as 1/2 are invisible), but inconsistency proves it wrong.
Rounding rules
× ÷ : fewest significant figures; + − : fewest decimal places
Conditions and limits: Apply to the final answer. Exact numbers (counted objects, defined conversions) do not limit precision.
Why it works: Why 1 m/s = 3.6 km/h
Convert 1 m/s to km/h with conversion factors.
1 m / s × 1 km / 1000 m × 3600 s / 1 h.
Arrange each factor so m and s cancel.
= 3600 / 1000 km / h = 3.6 km/h.
Multiply the numbers; the units left are km/h.
So to go from km/h to m/s divide by 3.6: 90 km/h = 25 m/s; 36 km/h = 10 m/s.
A first worked example
How to handle units in any calculation
- Write every given quantity with its unit, and convert all of them to SI (m, kg, s) before substituting.
- Build conversions as chains of factors equal to 1; check that unwanted units cancel.
- Before substituting numbers, check the dimensions of the formula you are using.
- Carry units through the calculation; the unit of the answer should appear automatically.
- Round the final answer to the appropriate number of significant figures, and ask whether its size is reasonable.
Speed conversion
Problem. Convert 36 km/h to m/s.
36 km / h × 1000 m / 1 km × 1 h / 3600 s.
km and h cancel.
= 36 000 / 3600 m/s = 10 m/s.
Multiply the numbers.
Result: 36 km/h = 10 m/s.
What it means: Quick check: divide by 3.6.
A different case
Density with a unit conversion
Problem. A sample has mass 250 g and volume 100 cm3. Find its density in g/cm3 and in kg/m3.
ρ = 250 g / 100 cm3 = 2.5 g/cm3.
Definition of density.
2.5 g / cm3 × 1 kg / 1000 g × 106 cm3 / 1 m3 = 2500 kg/m3.
1 m³ = 10⁶ cm³ (cube of 100 cm/m).
Result: ρ = 2.5 g/cm3 = 2.5 × 103 kg/m3.
What it means: About two and a half times as dense as water — typical of rock or aluminium (2.70 g/cm³).
More worked cases
Each case below uses a different skill. Every step and result is shown.
Dimensional check of an equation
Problem. A student writes v2 = v02 + 2at for motion with constant acceleration. Is it dimensionally correct?
Left side: (m/s)2 = m2/s2.
Dimensions of velocity squared.
Term 2at: (m/s2)(s) = m/s.
Acceleration times time is a velocity, not a velocity squared.
m2/s2 cannot be added to m/s.
Terms added together must have the same dimensions.
Result: The equation is wrong. The correct form is v2 = v02 + 2aΔx: (m/s²)(m) = m²/s².
What it means: A dimension check catches the error without any numbers.
Significant figures in a calculation
Problem. A block measures 2.45 cm × 3.1 cm × 1.20 cm. Report its volume.
V = 2.45 × 3.1 × 1.20 = 9.114 cm3 (unrounded).
Multiply; keep all digits for now.
The least precise factor, 3.1 cm, has 2 significant figures.
For multiplication, the fewest significant figures decide.
Result: V ≈ 9.1 cm3.
What it means: Reporting 9.114 cm³ would claim a precision the measurements do not have.
An area conversion
Problem. A plate has area 250 cm2. Express it in m2.
1 m = 100 cm, so 1 m2 = 1002 cm2 = 104 cm2.
Square the length factor.
250 cm2 × 1 m2 / 104 cm2 = 0.025 m2.
Arrange to cancel cm².
Result: 0.025 m2 (= 2.5 × 10−2 m2).
What it means: Using 100 instead of 10⁴ would give 2.5 m², a hundred times too large.
Common misunderstandings
Misunderstanding: Converting cm² to m² by dividing by 100.
Correct idea: Square the factor: divide by 10⁴. For volumes divide by 10⁶.
Misunderstanding: Mixing units in one formula (km with m, minutes with seconds).
Correct idea: Convert everything to SI first.
Misunderstanding: Rounding at every intermediate step.
Correct idea: Carry extra digits and round only the final answer, or rounding errors accumulate.
Misunderstanding: “Dimensionally consistent means correct.”
Correct idea: Consistency is necessary, not sufficient; numerical factors like ½ cannot be checked by dimensions.
Misunderstanding: Confusing mass (kg) and weight (N).
Correct idea: Mass is the amount of matter; weight is the gravitational force mg, measured in newtons.
Keep in mind
- 1 m/s = 3.6 km/h; v(m/s) = v(km/h) / 3.6Speed conversion
- 1 m2 = 104 cm2; 1 m3 = 106 cm3 = 103 L; 1 mL = 1 cm3Area and volume conversions
- ρ = m / VDensity
- [left side] = [right side]; only like dimensions can be addedDimensional consistency
- × ÷ : fewest significant figures; + − : fewest decimal placesRounding rules
Scope of this lesson
- Uncertainty propagation beyond significant-figure rules (standard deviations, combined uncertainties) is introduced in the laboratory topic and developed in laboratory courses.
Next: Vectors and motion. Describe motion with position, velocity and acceleration as vectors; with constant acceleration, four equations predict everything, and horizontal and vertical motions are independent.
Original study text. Sources and credits.